Question

for the triangles to be similar by the sss similarity theorem, what must be the value of y? 18 20 24 26

Explanation:

Step1: Recall SSS Similarity Theorem

The SSS (Side - Side - Side) similarity theorem states that if the corresponding sides of two triangles are in proportion, then the triangles are similar. So, we need to set up a proportion between the sides of the two triangles.

Let's denote the first triangle as $\triangle ABC$ with sides $AC = 14$, $BC=10$, $AB = 10$ and the second triangle as $\triangle PQR$ with sides $PR = 28$, $RQ=y$, $PQ = y$.

The ratio of corresponding sides should be equal. The ratio of $AC$ to $PR$ is $\frac{AC}{PR}=\frac{14}{28}=\frac{1}{2}$.

Step2: Set up Proportion for Other Sides

Since the triangles are similar by SSS, the ratio of $BC$ to $RQ$ (and also $AB$ to $PQ$) should be equal to the ratio of $AC$ to $PR$.

We know that $BC = 10$ and $RQ=y$, and the ratio $\frac{BC}{RQ}=\frac{10}{y}$. Also, $\frac{AC}{PR}=\frac{14}{28}=\frac{1}{2}$.

So we set up the proportion $\frac{10}{y}=\frac{1}{2}$? Wait, no, wait. Wait, maybe I mixed up the sides. Wait, let's re - examine the triangles.

Wait, in $\triangle ABC$, the sides are $AB = 10$, $BC = 10$, $AC=14$? Wait, no, looking at the diagram, $A$ to $B$ is 10, $B$ to $C$ is 10, $A$ to $C$ is 14. And in $\triangle PQR$, $P$ to $Q$ is $y$, $Q$ to $R$ is $y$, $P$ to $R$ is 28.

So the corresponding sides: $AB$ corresponds to $PQ$, $BC$ corresponds to $QR$, and $AC$ corresponds to $PR$.

So the ratio of similarity is $\frac{AC}{PR}=\frac{14}{28}=\frac{1}{2}$. Wait, no, that would mean the smaller triangle is $\triangle PQR$? Wait, no, $AC = 14$ and $PR=28$, so $\triangle ABC$ is the smaller triangle and $\triangle PQR$ is the larger one? Wait, no, 14 is less than 28, so the scale factor from $\triangle ABC$ to $\triangle PQR$ is $\frac{PR}{AC}=\frac{28}{14} = 2$.

So the length of $PQ$ (which corresponds to $AB$) should be $AB\times2$. Since $AB = 10$, then $PQ=y=10\times2 = 20$. Similarly, $QR$ (which corresponds to $BC$) should be $BC\times2=10\times2 = 20$, so $y = 20$.

Answer:

20